Rolle’s Theorem
if f(x) is continuous on [a,b] and differentiable on (a,b), and f(a)=f(b), then there is a number c between (a,b) where f'(c) = 0
Rolle’s Theorem is a fundamental theorem of calculus that deals with the existence of at least one point on a differentiable function’s graph where the tangent is horizontal. Specifically, Rolle’s Theorem states that if a function f(x) is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and if f(a) = f(b), then there exists at least one point c in the open interval (a, b) where the derivative of the function f(c) equals zero, and therefore the tangent of the function at that point is horizontal.
The theorem is named after Michel Rolle (1652-1719), a French mathematician who first stated and proved it in 1691. Rolle’s Theorem is an essential tool in calculus and is used to prove other theorems related to a function’s behavior on a closed interval.
To better understand the theorem, consider the graph of a continuous, differentiable function f(x) over the interval [a, b], such that f(a) = f(b). If the function has a local maximum or minimum value within that interval, then the derivative of f(x) at that point must be zero. This is because the slope of the tangent line at that point is zero, and the tangent line is horizontal. Therefore, there must be at least one point c in the open interval (a, b) with f'(c) = 0.
In summary, Rolle’s Theorem states that a differentiable function that has the same values at the endpoints of a closed interval must have at least one horizontal tangent line between those endpoints.
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